LINEAR PROGRAMMING
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BOARD PROBLES EXERCISE – II
A producer has 30 and 17 units of labour and capital respectively which he can use to produce two type of goods X and Y. To produce one unit of X, 2 units of labour and 3 units of capital are required. Similarly, 3 units of labour and 1 unit of capital is required to produce one unit of Y. If X and Y are priced at Rs 100 and 120 per unit respectively, how should the producer use his resources to maximise the total revenue ? Solve the problem graphically. [C.B.S.E. 2000] A company manufactures two types of toys A and B. Type A requires 5 minutes each for cutting and 10 minutes each for assembling. Type B requires 8 minutes each for cutting and 8 minutes each for assembling. There as 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs 50 each on type A and Rs 60 each on type B. How many toys of each type should the company manufacture in a day to maximise the profit ? [C.B.S.E. 2001] A company sells two different products A and B. The two products are produced in a common production process which has a total capacity of 500 man hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The demand in the market shows, that the maximum number of units of A that can be sold is 70 and that for B is 125. Profit on each unit of A is Rs 20 and that on B is Rs 15. How many units of A and B should be produced to aximise the profit ? Solve it graphically. [C.B.S.E. 2003] A manufacturer makes two types of cups, A and B. Three machines are required to manufacture the cups and the time in minutes required by each is as given below :
Type of Cup A B
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I 12 6
Machines II 18 0
III 6 9
Each machine is available for a maximum period of 6 hours per day. If the profit on each cup A is 75 paise, and on B it is 50 paise, show that 15 cups of type A and 30 cups of type B should be manufactured per day to get the maximum profit. [C.B.S.E. 2003] Kellogg is a new cereal formed of a mixture of bran and rice that contains atleast 88 grams of protein and atleast 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing the new cereal if bran costs Rs 4 per kilogram and rice costs Rs 4 per kilogram. [C.B.S.E. 2003] A man has Rs 1500 for purchase of rice and wheat. A bag of rice and a bag of wheat cost Rs 180 and Rs 120 respectively. He has a storage capacity of 10 bags only. He earns a profit of Rs 11 and Rs 9 respectively per bag of rice and wheat. Formulate it as a linear programming problem and solve it graphically for maximum profit. [C.B.S.E. 2004] Anil wants to invest atmost Rs 12000 in Bonds A and B. According to rules, he has to invest atleast Rs 2000 in Bond A and atleast Rs 4000 in Bond B. If the rate of interest on Bond A is 8% p.a. and on Bond B is 10% p.a., how should he invest his money for maximum interest ? Solve the problem graphically. [C.B.S.E. 2004] A firm deals with the two kinds of fruit juices–pine apple and orange juice. These are mixed and two mixtures are sold are soft drinks A and B. One tin of A requires 4 litres of pine apple and 1 litre of orange juice. One tin of B requires 2 litres of pine apple and 3 litres of orange juice. The firm has only 46 litres of pine apple juice and 24 litres of orange juice. Each tin of A and B are sold at a profit of Rs 4 and Rs 3 respectively. How many tins of each type should the firm produce to maximise the profit ? Solve the problem graphically. [C.B.S.E. 2004] A manufacturer makes two products, A and B. Product A sells at Rs 200 per unit and takes 30 minutes to make. Product B sells at Rs 300 per unit and takes 1 hour to make. There is a permanent order of 14 units of product A and 16 units of product B. A working week consists of 40 hours of production and the weekly turn over must not be less than Rs 10000. If the profit on each of product A is Rs 20 and on product B is Rs 30, then how many of each should be produced so that the profit is maximum ? Also, find the maximum profit. Solve the problem graphically. [C.B.S.E. 2005] Two tailors A and B earn Rs 15 and Rs 20 per day respectively. A can stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce atleast 60 shirts and 32 pants at a minimum labour cost ? Solve the problem graphically. [C.B.S.E. 2005] A farmer has a supply of chemical fertilizers of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which contains 5% nitrogen and 10% of phosphoric acid. After soil testing it is found that atleast 7 kg of nitrogen and the same quantity of phosphoric acid is required for a good crop. The fertilizer of type A costs Rs 5 per kg and the type B costs Rs 8 per kg. Using linear programming find how many kilograms of each type of the fertilizer should be bought to meet the requirement and the cost be minimum. Solve the problem graphically. [C.B.S.E. 2006]
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LINEAR PROGRAMMING A furniture dealer deals only in two items – tables and chairs. He has Rs 10000 to invest and a space to store atmost 60 pieces. A table costs him Rs 500 and a chair Rs 200. He can sell a table at a profit of Rs 50 and a chair at a profit of Rs 15. Assume that he can sell all items that he buys. Using linear programming formulate the problem for maximum profit and solve it graphically. [C.B.S.E. 2006] A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5760 to invest and has space for atmost 20 items. A fan and sewing machine cost Rs 360 and Rs 240 respectively. He can sell a fan at a profit of Rs 22 and sewing machine at a profit of Rs 18. Assuming that he can sell whatever he buys, how should he invest his money in order to maximise his profit ? Translate the problem into L.P.P and solve it graphically. [C.B.S.E. 2006] If a young man rides his motorcycle at 25 km/h, he had to spend Rs 2 per km on petrol. If he rides at a faster speed of 40 km/h, the petrol cost increase at Rs 5 per km. He has Rs 100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this as on L.P.P and solve it graphically. [C.B.S.E. 2007] An aeroplane can carry a maximum of 200 passengers. A profit of Rs 400 is made on each first class ticket and a profit of Rs 300 is made on each second class ticket. The airline reserves atleast 20 seats for first-class. However, tickets of each type must be sold to maximise profit for the airline. Form an LPP and solve it graphically. [C.B.S.E. 2008] A farmer has a supply of chemical fertilizer of type A which contains 10% nitrogen and 5% phosphoric acid, and type B which contains 6% nitrogen and 10% phosphoric acid. After testing the soil conditions of the field, it was found that at least 14 kg of nitrogen and 14 kg of phosphoric acid is required for producing a good crop. The fertilizer of type of A costs Rs 5 per kg and the type B costs Rs 3 per kg. How many kg of each type of the fertilizer should be used to meet the requirement at the minimum possible cost ? Using LPP, solve the above problem graphically. [C.B.S.E. 2008] A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 units of calories. Two foods A and B are available at a cost of Rs 5 and Rs 4 per unit respectively. One unit of the food A contains 200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of the food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost but it must satisfy the requirements of the sick person. Form the question as LPP and solve it graphically.[C.B.S.E. 2008] One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the above as a linear programming problem and solve graphically. [C.B.S.E. 2009] A diet is to contain 80 units of vitamin A and 100 units of minerals. Two foods F 1 and F2 are available. Food F1 costs Rs 4 per unit and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meet the minimal nutritional requirements. [C.B.S.E. 2009] A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day 7 is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number hours available per day is 16. If the profit on a ring is Rs 300 and that on a chain is Rs 190, find the number of rings and chains that should be manufactured per day, so as the earn the maximum profit. Make it as an L.P.P. and solve it graphically. [C.B.S.E. 2010] A merchant plans to sell two types of personal computers – a desktop model and a protable model that will cost Rs. 25,000 and Rs. 40,000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if the does not want to invest more than Rs. 70 lakhs and his profit on the desktop model is Rs. 4,500 and on the portable model is Rs. 5,000. Make an L.P.P. and solve it graphically. [C.B.S.E. 2011] A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 units/kg of vitamin A and 2 units /kg of vitamin C. It costs Rs. 5 per kg to purchase Food I and Rs. 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically. [C.B.S.E. 2012] A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs. 100 and Rs. 120 per unit respectively, how should he use his resources to maximise the total revenue ? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate ? [C.B.S.E. 2013]
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LINEAR PROGRAMMING
164
ANSWER KEY EXERCISE – 1 (UNSOLVED PROBLEMS) 1. The maximum value of z = <
570 15 , when x = and y = 4 7 7
2. 10 rings and 20 chains ; maximum profit = Rs 5500
3. Good X = 3 units, good Y = 8 units ; maximum revenue = Rs 1260 4. Product A = 60 units, product B = 240 units ; maximum profit = Rs 1020 5. Food X = 6 units, Food Y = 6 units ; minimum cost = Rs 300 6. No minimum value 7. Fertilizer type 1 = 100 kg, fertilizer type II = 80 kg ; minimum cost = Rs 92 8. Type A = 20 trees, type B = 40 trees ; maximum profit = Rs 3200 9. First class = 40 tickets, economy class = 160 tickets ; maximum profit = Rs 64,000 10. Type A = 6 machines, type B = 0 machines ; maximum output = 360 units 11. Type I = 0 trunks, type II = 6 trunks; maximum profit = Rs 150. 12. Type A =800 dolls, type B = 400 dolls; maximum profit, Rs 16000 13. Brand P = 40 bags, brand Q = 100 bags ; minimum amount of nitrogen = 470 kg 14. Crop X = 30 hectares, Crop Y = 20 hectare ; maximum profit = Rs. 4,95,000 15. Minimum cost = Rs 400
Factory A Factory B
No. of backets transported Agency P Agency Q Agency R 10 0 50 30 40 0
EXERCISE – 2 (BOARD PROBLEMS)
1. 1260, (3, 8) 2. 1500, (12, 15) 3. A : 25 units, B : 125 units 6. 100, (5, 5) 7. (2000, 10000) 8. A : 9 tin, B : 5 tin 9. 1440, (48, 16)
5. Rs 4.6, (.6,.4) 10. A : 5 days, B : 3 days
50 40 16. A : 80 kg, B : 100 Kg 11. 570, (50, 40) 12. 1000, (20, 0) 13. 392, (8, 12) 14. 30, , 3 3 17. A : 5 unit, B : 30 unit 18. 30 19. 104 20. 5440, (8, 16) 21. 1150000, (200, 50) 22. 38, (2, 4)
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